What is the factored form of the expression 2x^2 - 7x + 3?
Understand the Problem
The question is asking for the factored form of the quadratic expression 2x^2 - 7x + 3, and provides multiple choice options for the answer. We will need to factor the expression to find the correct option.
Answer
The factored form is \( (2x - 1)(x - 3) \).
Answer for screen readers
The factored form of the quadratic expression ( 2x^2 - 7x + 3 ) is ( (2x - 1)(x - 3) ).
Steps to Solve
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Identify the Quadratic Expression The expression we want to factor is ( 2x^2 - 7x + 3 ).
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Multiply the Leading Coefficient and Constant Term Multiply the leading coefficient (2) by the constant term (3): $$ 2 \times 3 = 6 $$
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Find Factor Pairs of the Result We need to find two numbers that multiply to 6 and add to the middle coefficient (-7). The numbers are (-6) and (-1) because: $$ -6 \times -1 = 6 \quad \text{and} \quad -6 + -1 = -7 $$
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Rewrite the Middle Term Now, we can rewrite the middle term (-7x) using the two numbers found: $$ 2x^2 - 6x - 1x + 3 $$
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Group the Terms Next, we will group the terms: $$ (2x^2 - 6x) + (-1x + 3) $$
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Factor by Grouping Factor out common factors in each group: $$ 2x(x - 3) - 1(x - 3) $$
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Factor Out the Common Binomial Now factor out the common binomial factor ((x - 3)): $$ (2x - 1)(x - 3) $$
The factored form of the quadratic expression ( 2x^2 - 7x + 3 ) is ( (2x - 1)(x - 3) ).
More Information
Factoring quadratics is a common algebraic process that helps simplify expressions or solve for roots. The method of grouping is a useful technique to factor polynomials effectively.
Tips
- Incorrectly Finding Factor Pairs: Make sure the numbers you choose multiply to the product of the leading coefficient and the constant and add to the middle coefficient.
- Forgetting to Factor Out Completely: Always check to factor out any common factors from each grouping section.
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